9. Suppose a point (X,Y) is selected at random from inside the circle with radius 2...
A point is selected randomly inside a circle with a radius of R. X denotes the distance between the selected position to circle's center. First, calculate the probability function of X, then find the best results below for the P( R/3 < X < R/2). (Hint: First you should calculate the distribution function of X). Select one: a. 4/19 b. 12/41 c. 4/23 d. 5/36
Suppose that a point X is selected at random from the interval (0,1). After the value X = x has been selected, a point Y is then chosen at random from the interval (0,x^2). a) Indicate the region R on the xy-plane of possible values of the random vector (X,Y). b) Find the marginal pdf f2(y) of Y.
Find the conditional p.d.f.’s f(y|x) and f(z|x, y).
4. Suppose that random variables (X, Y, Z) have the joint p.d.f. f(x,y,z)-' 0, otherwise . ind the conditional p.d.f.'s f(yx) and f (z x,y
A circular plate of radius 4 is heated. The temperature at point (x, y) on the plate is given by f(x, y) = 2x 2 + 3y 2 − 4x + 5. Assume (0,0) is the center of the plate. (a) Find the hottest and coolest points on the edge of the plate. (b) Is there a point inside the disc that is hotter? Is there a point that’s cooler?
5. * Suppose that a point is chosen at random in the interior of a circle of radius 1. Let D be the distance of the selected point from the centre of the circle. What is the density function of D?
10. (calculations with independent Gaussians) The joint pdf of two random variables is given by fxy(x,y) = [2끼-iexp[-2(z2 + y2)] for-x 〈 x, y 〈 oo. Compute the probability that both X and Y are restricted to (a) the 2 x 2 square, where 1 < r,y 3 1; and (b) the unit circle, which has its center at the origin (0,0) with a radius of 1
6. (extra credit) Find the center of mass of a region inside a circle of radius a if the density at any point is proportional to its distance from the center. (Either compute the center, or guess it and give a theoretical argument why your guess is correct.)
6. (extra credit) Find the center of mass of a region inside a circle of radius a if the density at any point is proportional to its distance from the center. (Either...
Suppose X, Y are random variables whose joint PDF is given by fxy(x, y) 9 { 0 <y <1,0 < x <y y otherwise 0, 1. Find the covariance of X and Y. 2. Compute Var(X) and Var(Y). 3. Calculate p(X,Y).
QUESTION 9 Let the random variable X and Y have the joint p.d.f. f(x,y) for the (x,y) pairs as shown in the following table (for x = 0,1,2 and y = 0.1). y/X 0 1 2 0 1 14 6 | 18 18 1133 18 18 Find the covariance oxy O-57/324 O-58/324 57/324 58/324
السر و عقد الأجبية The level curve of f(x,y)= are 25-x- A circle center at (0,0) and radius. IVO 25 - .: CE(-0,0) A circle center at (0) and radius III 25 .: CE(-0, A circe center at (0,0) and radius 25 - : C 25-3: cela) A circle center at (0) and radius lo 25 -: CE- celestes